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Formal Connections for Families of Star Products

Abstract

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection, which was introduced by the first author. We establish a necessary and sufficient condition that guarantees the existence of a formal connection, and we describe the space of formal connections for a family as an affine space modelled on the formal symplectic vector fields. Moreover, we showthat if the parameter space has trivial first cohomology group, any two flat formal connections are related by an automorphism of the family of star products.

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Correspondence to Jørgen Ellegaard Andersen.

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This work was partially supported by the center of excellence grant ‘Centre for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95).

Communicated by N. Reshetikhin

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Andersen, J.E., Masulli, P. & Schätz, F. Formal Connections for Families of Star Products. Commun. Math. Phys. 342, 739–768 (2016). https://doi.org/10.1007/s00220-016-2574-2

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Keywords

  • Modulus Space
  • Toeplitz Operator
  • Symplectic Manifold
  • Formal Power Series
  • Star Product